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Beauty in Mathematical Proofs

Research project School mathematics, for a variety of reasons, has little resemblance to what mathematicians actually do in their daily lives. Many students experience mathematics as routine and perhaps even boring, whereas mathematicians describe their work in terms of beauty and play.

While it is not at all trivial to bring the practice of mathematics into the mathematics classroom, and perhaps even questionable if this is the best way to educate all students, it seems a shame for students to go through their entire mathematics education without really getting a feel for what the subject is like. As it is now, many talented students choose to end their studies in mathematics without experiencing real mathematics, and many students who struggle with the subject will do so without experiencing any of the rewards of doing so. This study on beauty, and the subsequent work we hope to do to connect our results to practices of teaching and learning, have the aim of providing students an opportunity to experience real mathematics along the lines of how mathematics is actually practiced, not just as a utilitarian subject, but a human one.

Head of project

Manya Sundström
Research fellow, associate professor
E-mail
Email

Project overview

Project period:

2011-04-29 2013-12-31

Participating departments and units at Umeå University

Department of Science and Mathematics Education, Faculty of Science and Technology

Research area

Educational sciences, Mathematics

Project description

Many people throughout history have claimed that mathematics is beautiful. For instance the poet Edna St. Vincent Millay famously wrote, ”Euclid alone has looked on beauty bare.” And G.H. Hardy, a prominent mathematician from the early 20th century wrote, ” The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way ” However we know less about why mathematics appears to us as beautiful. This question seems important to answer, both to better understand the nature of beauty in mathematics and to inform discussions about how beauty might be used in the teaching of mathematics.

While beauty has been discussed since ancient times, we have surprisingly little empirical data of mathematicians making judgments about beauty, which is what this project seeks to provide. The product of this project will be approximately eight case studies of mathematicians from different fields describing the ways in which particular proofs are or are not beautiful. Results from a pilot study indicate that two potential features which inform judgments of beauty are generality and specificity. Generality refers to the fact that a particular proof might be one instance of a more general result. The connection to this general result lends the proof a sort of status and a feeling of ”Oh, that theorem applies even here!”. Specificity refers to the fact that in order to generate a proof, one sometimes needs to use an argument specific to that situation in order to reveal a type of hidden or unexpected result. One example is a proof of Pick’s theorem in which an idea of counting up angle measures of triangles in different ways to gives information about the area of a bounding polygon. Pilot studies have indicated that both generality and specificity can lead to a judgment that a particular proof is beautiful.

Keywords: Proof, beauty, aesthetics, motivation, mathematics education